Question

two sets of data are shown. data set a: 30, 38, 42, 42, 43, 47, 51, 51, 57, 59 data set b: 38, 39, 40, 42, 44, 46, 47, 50, 51, 52 choose all the measures which are greater for data set a than for data set b. mean range median standard deviation interquartile range

Explanation:

Step1: Calculate the mean of Data Set A

$\text{Mean}_A=\frac{30 + 38+42+42+43+47+51+51+57+59}{10}=\frac{460}{10} = 46$

Step2: Calculate the mean of Data Set B

$\text{Mean}_B=\frac{38+39+40+42+44+46+47+50+51+52}{10}=\frac{451}{10}=45.1$

Step3: Calculate the range of Data Set A

$\text{Range}_A=59 - 30=29$

Step4: Calculate the range of Data Set B

$\text{Range}_B=52 - 38 = 14$

Step5: Calculate the median of Data Set A

Since $n = 10$ (even), $\text{Median}_A=\frac{43 + 47}{2}=45$

Step6: Calculate the median of Data Set B

Since $n = 10$ (even), $\text{Median}_B=\frac{44+46}{2}=45$

Step7: Calculate the standard - deviation of Data Set A

First, find the variance.
$\text{Variance}_A=\frac{\sum_{i = 1}^{n}(x_i-\text{Mean}_A)^2}{n}$
$=\frac{(30 - 46)^2+(38 - 46)^2+(42 - 46)^2+(42 - 46)^2+(43 - 46)^2+(47 - 46)^2+(51 - 46)^2+(51 - 46)^2+(57 - 46)^2+(59 - 46)^2}{10}$
$=\frac{256+64 + 16+16+9+1+25+25+121+169}{10}=\frac{692}{10}=69.2$
$\text{SD}_A=\sqrt{69.2}\approx8.32$

Step8: Calculate the standard - deviation of Data Set B

First, find the variance.
$\text{Variance}_B=\frac{\sum_{i = 1}^{n}(x_i-\text{Mean}_B)^2}{n}$
$=\frac{(38 - 45.1)^2+(39 - 45.1)^2+(40 - 45.1)^2+(42 - 45.1)^2+(44 - 45.1)^2+(46 - 45.1)^2+(47 - 45.1)^2+(50 - 45.1)^2+(51 - 45.1)^2+(52 - 45.1)^2}{10}$
$=\frac{50.41+37.21+26.01+9.61+1.21+0.81+3.61+24.01+34.81+47.61}{10}=\frac{235.3}{10}=23.53$
$\text{SD}_B=\sqrt{23.53}\approx4.85$

Step9: Calculate the inter - quartile range of Data Set A

First, find the first quartile $Q_1$ and third quartile $Q_3$.
The ordered data set is $30,38,42,42,43,47,51,51,57,59$.
$Q_1$ is the median of the lower half. The lower half is $30,38,42,42,43$, so $Q_1 = 42$.
$Q_3$ is the median of the upper half. The upper half is $47,51,51,57,59$, so $Q_3 = 51$.
$\text{IQR}_A=Q_3 - Q_1=51 - 42 = 9$

Step10: Calculate the inter - quartile range of Data Set B

The ordered data set is $38,39,40,42,44,46,47,50,51,52$.
The lower half is $38,39,40,42,44$, so $Q_1 = 40$.
The upper half is $46,47,50,51,52$, so $Q_3 = 50$.
$\text{IQR}_B=Q_3 - Q_1=50 - 40 = 10$

Answer:

mean, range, standard deviation